The counterexample proves that the converse of the theorem is not always true, and we have to be careful when making statements about limit points and convergence of sequences.
A sequence is a set of numbers that are generated one after the other in a specific pattern.
The theorem states that if a sequence converges, it will have exactly one limit point. In other words, if a sequence is getting closer and closer to a certain number, there will be only one number that it approaches. This is a fundamental idea in mathematics, and it is important to understand why it is true.
To prove this theorem, let's consider a sequence (a_n) that converges to a limit L. If there were two different limit points, say L1 and L2, then there would be a contradiction. This is because by definition, the sequence (a_n) would be getting closer and closer to L1 and L2 at the same time, which is not possible. Hence, it must be true that the sequence (a_n) has exactly one limit point, L.
The converse of this theorem is not always true. In other words, just because a sequence has exactly one limit point, it doesn't necessarily mean that it converges to that limit. To see why this is the case, let's consider the following sequence:
(2, 3, 4, 5, 6, ...)
In this sequence, the limit point is 6. However, the sequence does not converge to 6 because it is not getting closer and closer to 6 as it goes on. Instead, it is increasing by 1 with each term, and therefore it does not converge.
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In the figure below, m1 = (x+84)° and m2 = 2x°.
Find the angle measures.
Answer: The angle measures cannot be determined from the information given. The equation m1 = (x+84)° and m2 = 2x° only specify a relationship between two angles, but do not specify specific angle measures. To determine the angle measures, additional information or constraints are needed.
Step-by-step explanation:
Write the augmented matrix for the following system of equations 7-6y+8x=0, 6x=-3-2y
Augmented matrix is a matrix that is composed of the coefficients of the variables of a system of linear equations and the constants on the right side of the equal signs of the linear equation.
For the given system of equations, the augmented matrix is:
| 7 -6 8 | 0 |
| 6 -2 -3 | -3 |
The first row consists of the coefficients of the variables x and y, and the constant 0 from the equation 7-6y+8x=0.
The second row consists of the coefficients of the variables x and y, and the constant -3 from the equation 6x=-3-2y.
The augmented matrix can be written in a short form as:
| 7 -6 8 | 0 |
| 6 -2 -3 | -3 |
This augmented matrix can be used to solve the system of equations by using the Gaussian Elimination Method or the Gauss-Jordan Elimination Method.
The columns of two matrices are combined to create a new matrix, which is known as an augmented matrix. When using matrices to solve straightforward linear equations, the augmented matrix is a crucial tool. The number of variables in the linear equation is the same as the number of rows in the augmented matrix.
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The rectangular floor of a classroom is 32 feet in length and 28 feet in width. A scale drawing of the floor has a length of 16 inches. What is the perimeter, in inches, of the floor in the scale drawing?
The perimeter of the floor in the scale drawing is 30 inches.
The ratio of length and width of floor of classroom will be same as the ratio of scale drawing. Therefore, representing the formula for ratio -
length/width (floor) = length/width (scale drawing)
32/28 = 16/width
Converting the original measurement to inches
1 foot = 12 inches
Length = 384 inches
Width = 336 inches
Rearranging the equation according to width of scale drawing
Width = (16 × 336)/384
Performing multiplication and division on Right Hand Side of the equation
Width = 14 inches
Thus, the width is 14 inches
Now, perimeter = 2(length + width)
Perimeter = 2 (16 + 14)
Perimeter = 2 × 30
Perimeter = 60 inches
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Pocahontas takes a sheet of paper and cuts from one corner to the opposite corner, making two triangles. If the piece of paper is 12 inches long and 3.5 inches wide, what is the perimeter of one of the triangles? Provide an answer accurate to the nearest tenth.
The perimeter of one of the triangles is 28 inches.
What is Pythagoras's theorem?In a right-angled triangle, the sum of the squares of the smaller two sides of a right-angle triangle is equal to the square of the largest side.
From the given information the diagonal of the rectangle will be,
d = √(12² + 3.5²)
d = √(144 + 12.25).
d = √156.25.
d = 12.5.
We know, Perimeter is the sum of the lengths of all the sides.
Therefore, P = 12.5 + 12 + 3.5 inches.
P = 28 inches.
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The students of three sections of a class have to stand n rows. Each row have equal number of students. If there are 36, 40 and 48 students are in three sections. Find the maximum number of students in each rows.
Answer:
24
Step-by-step explanation:
The total number of students in the three sections is 36 + 40 + 48 = 124.
To find the maximum number of students in each row, we need to divide the total number of students by the number of rows, rounded down to the nearest integer.
Let's call the maximum number of students in each row "x". Then, we can write the equation:
x * n = 124
To find the maximum value of x, we need to find the maximum value of n such that the result of x * n is less than or equal to 124.
Starting with n = 1, we can increment n and calculate x for each value of n until x * n is greater than 124.
For n = 1, x * n = 124, so x = 124.
For n = 2, x * n = 62, so x = 62.
For n = 3, x * n = 41, so x = 41.
For n = 4, x * n = 31, so x = 31.
For n = 5, x * n = 24.8, which is not an integer, so we round down to 24.
Therefore, the maximum number of students in each row is 24, with a total of 5 rows.
Answer:
Step-by-step explanation:
Let's call the maximum number of students in each row "r". To find this value, we need to divide the total number of students by the number of rows. To ensure that all the students can fit into the rows, the number of students in each section must be divisible by the number of rows.
First, we find the least common multiple (LCM) of the number of students in each section, which will be the total number of students if we arrange them in the same number of rows. The LCM of 36, 40, and 48 is 240.
Next, we divide the LCM by the number of students in each section to find the number of rows:
r = LCM / 36 = 240 / 36= 6.67 (rounded down to 6)
So, the maximum number of students that can be in each row is 24
PLEASE HELP
Which of the following are solutions to the equation below?
CHECK ALL THAT APPLY.
(image attched
The solutions to the quadratic equation are;
[tex]x = \frac{ \sqrt{10} + 5 }{2} [/tex]
[tex]x = \frac{ - \sqrt{10} + 5 }{2} [/tex]
The correct answer choice is option A and E
How to solve quadratic equation using formula?4x² - 20x + 25 = 10
Equate the equation to 0
4x² - 20x + 25 - 10 = 0
4x² - 20x + 15 = 0
Solve the quadratic equation using formula
[tex]4x^2 - 20x + 15 = 0[/tex]
using Quadratic Formula where,
a = 4,
b = -20, and
c = 15
[tex]x = \frac{ -b \pm \sqrt{b^2 - 4ac}}{ 2a }[/tex]
[tex]x = \frac{ -(-20) \pm \sqrt{(-20)^2 - 4(4)(15)}}{ 2(4) }[/tex]
[tex]x = \frac{ 20 \pm \sqrt{400 - 240}}{ 8 }[/tex]
[tex]x = \frac{ 20 \pm \sqrt{160}}{ 8 }[/tex]
[tex]x = \frac{ 20 \pm 4\sqrt{10}\, }{ 8 }[/tex]
[tex]x = \frac{ 20 }{ 8 } \pm \frac{4\sqrt{10}\, }{ 8 }[/tex]
[tex]x = \frac{ 5}{ 2 } \pm \frac{ \sqrt{10}\, }{ 2 }[/tex]
Therefore, the solution to the quadratic equation using formula is
[tex]x = \frac{ 5}{ 2 } \pm \frac{ \sqrt{10}\, }{ 2 }[/tex]
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Express the given equation in standard form. Identify the values of a, b, and c.
3x = -x²+7
Evaluate 2 over 3 times 6 minus 3 plus 5 over 3 times 3
Answer: 2/21
Step-by-step explanation:
2/3x6-3+5/3x3
2/18-3+5
2/21
Tenesha is trying to draw a parallelogram and find its area using the two line segments shown.
On a coordinate plane, 2 line segments are formed by points (1, 5), (5, 2), (12, 2).
She begins by finding the missing vertex on the graph to be point (9, 5). Then she calculates the area of the parallelogram. What errors did she make? Check all that apply.
On a coordinate plane, a parallelogram is formed by the points (1, 5), (5, 2), (12, 2). and fourth point unlabeled. A = (8) (5). A = 40.
She did not find the correct vertex; it should be at (8, 5).
She used the wrong area formula; it should be A = one-half b h.
She used the wrong length for the base; it should be 7, like the given base.
She used a side length of the parallelogram for the height instead of a line segment perpendicular to the base; the height should be 3.
She should have found a segment for the top that was not parallel to the bottom.
The errors she made are She did not find the correct vertex; it should be at (8, 5) and She used the wrong area formula; it should be A = one-half b h, the incorrect options are B and C.
What is standard error?Error is an estimate of the standard deviation of the sampling distribution. It measures the variability of a considered sample statistic.
Supopse that we're given that:
Population standard deviation =[tex]\sigma[/tex]
Size of sample we're working on = n
Then, the standard error can be calculated as:
[tex]SE = \dfrac{\sigma}{\sqrt{n}}[/tex]
where SE denotes the standard error.
Given;
On a coordinate plane, 2 line segments are formed by points (1, 5), (5, 2), (12, 2).
She begins by finding the missing vertex on the graph to be point (9, 5).
Now,
The parallelogram is a simple quadrilateral with two parallel pairs of sides; The opposite or reverse sides of the parallelogram are of similar duration and the reverse angles of the parallelogram are of equivalent proportion;
The square has two sets of parallel sides, four right angles, and the four sides are identical. It's even a triangle and a parallelogram;
The rhombus is known as a four-sided parallelogram;
Therefore, the error will be in B and C.
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solve the ineaquality. 5t+4≥19
First, subtract 4 from both sides:
[tex]5t\geq 15[/tex]
Now, divide both sides of the equation by 5 to isolate variable t:
[tex]t\geq 3[/tex]
Therefore, we know that t must be greater than or equal to 3 for this inequality to be true!
Solve the problem.
The profit made when t units are sold, t > 0, is given by P-t2-30t+200. Determine the number of units to be sold in
order for P> 0 (a profit is made).
A) 20
B) t=30
t = 20 ort=10
(D) t>20 or t < 10
The number of units sold for a profit to be made is given as follows:
D. t > 20 or t < 10.
How to make a profit?The quadratic function that models the earnings is given as follows:
P(t) = t² - 30t + 200.
The roots of the quadratic function are obtained as follows:
t² - 30t + 200 = 0
(t - 20)(t - 10) = 0.
Hence:
t - 20 = 0 -> t = 20.t - 10 = 0 -> t = 10.A profit is made when P(t) > 0, and since P(t) is a concave up function, it is positive to the left of the smallest root and to the right of the largest root, hence the correct option is given by option D.
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Eric and Jennifer both plan to run for a spot on the school board in their city. They must each collect a certain number of signatures to get their name on the ballot. So far, Eric has 20 signatures, but Jennifer just started and doesn't have any yet. Eric is collecting signatures at an average rate of 17 per hour, while Jennifer can get 19 signatures every hour. Assuming that their rate of collection stays the same, eventually the two will have collected the same number of signatures. How many signatures will they both have? How many hours will have gone by?
After 20 hours, Eric and Jennifer will each have 340 signatures.
We may begin by constructing an equation that solves for the number of signatures and the number of hours that have passed. Let's name the number of signatures "x" and the number of hours "h" respectively. Then there's:
20 + 17h Eric's signatures
Jennifer's signatures: 19h
We may put their signatures equal to each other because they will eventually have the same amount of signatures:
20 + 17h = 19h
We can now solve for h by removing h from one side of the equation. To begin, remove 19h from both sides:
20 = -h
Then we divide both sides by -1 to get h on its own:
h = 20
So, after 20 hours, Eric and Jennifer will have both gathered the same amount of signatures, which is:
Eric's autographs total 340 (20 + 17 * 20).
Jennifer has 19 * 20 = 380 signatures.
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2/5+1/3= using 15 as the  common denominator 
Answer:
11/15
Step-by-step explanation:
stu dent has 8 different math books, 9 different science books and 14 different engineering books. find the number of ways to pick two books with different subjects.
The total number of ways to select two books which are of different subject from the given subjects books is equal to 310.
Total number of different Math's books are 8
Total number of different Science books are 9
Total number of different Engineering books are 14
Number of ways to select two books with different subject
= (One Math's One Science ) + (One Math's One Engineering ) +(One Engineering One Science )
= ⁸C₁⁹C₁ + ⁸C₁¹⁴C₁ + ⁹C₁¹⁴C₁
= [ 8! / (8 -1)!1! ] [ 9! / (9 -1)!1! ] + [ 8! / (8 -1)!1! ] [ 14! / (14 -1)!1! ] + [ 9! / (9 -1)!1! ] [ 14! / (14 -1)!1! ]
= (8 × 9) + ( 8 × 14 ) + ( 9 × 14 )
= 72 + 112 + 126
= 310
Therefore, the number of ways to select two books of different subjects are 310.
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You are a travel agent and wish to estimate, with a 95% confidence, the proportion of vacationers who plan to travel outside the United States in the next 12 months. Your estimate must be accurate within 3% of the true proportion. a) no preliminary estimate is available. Find the minimum sample size needed. b) find the minimum sample size needed, using a prior study that found that 26% of the respondents said they planned to travel outside the United States in the next 12 months.
The minimum sample size needed is 193.
What do you mean by preliminary estimate?A preliminary estimate, also known as a preliminary budget or a rough estimate, is an early approximation of the cost or resources required for a project, program, or activity. It is made before a more detailed and accurate assessment is completed and is used to help make initial decisions and allocate resources.
A preliminary estimate is typically made based on available information and expert judgment, and it can be subject to revision as more information becomes available. It is an important tool for project planning, as it helps to establish a baseline for the cost or resource requirements of a project and provides a basis for making decisions about the feasibility and scope of the project.
A preliminary estimate can also be used to secure funding, establish project schedules, and identify potential risks and challenges. It is important to note that a preliminary estimate is not a final or binding commitment, and that the actual cost or resource requirements of a project may differ from the preliminary estimate.
a) No preliminary estimate is available:
The formula for minimum sample size for estimating a proportion with a margin of error and a given level of confidence is:
n = (Z² × p × (1 - p)) / E²
Where:
Z = the Z-score that corresponds to the desired level of confidence (95% confidence corresponds to a Z-score of 1.96)
p = estimated proportion (since no preliminary estimate is available, a commonly used value is 0.5)
E = desired margin of error (3%)
Plugging in the values, we get:
n = (1.96² × 0.5 × (1 - 0.5)) / (0.03²) = 384
So, the minimum sample size needed is 384.
b) Using a prior study that found 26% of the respondents said they planned to travel outside the US:
Plugging in the prior estimate of p = 0.26 and the same Z and E values as above, we get:
n = (1.96² × 0.26 × (1 - 0.26)) / (0.03²) = 193
So, the minimum sample size needed is 193.
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SHOW WORK A snowboarder stands at the edge of a 4.50 m high half-pipe. The mass of the snowboarder is 55.0 kg. What is the gravitational potential energy of the snowboarder thanks
Answer:
Step-by-step explanation:
The gravitational potential energy (GPE) of an object with mass m at a height h above some reference level is given by the formula:
GPE = mgh
where g is the acceleration due to gravity (which is approximately 9.8 m/s^2 on Earth).
In this case, the snowboarder has a mass of 55.0 kg and is at a height of 4.50 m above some reference level (which is not specified). So the GPE of the snowboarder is:
GPE = (55.0 kg) x (9.8 m/s^2) x (4.50 m) = 2404.5 J
Therefore, the gravitational potential energy of the snowboarder is 2404.5 joules (J).
Answer:
2404.5
Step-by-step explanation:
(3.18 + 5.7)/(0.3 * 5.6)
Answer:5.28571428571
Step-by-step explanation:
Answer:
(3.18 + 5.7)/(0.3 * 5.6) is approximately equal to 5.29.
Step-by-step explanation:
Following the order of operations, first we must calculate the values inside the parenthesis. The numerator will become 3.18 + 5.7 = 8.88, and the denominator will become 0.3 * 5.6 = 1.68. Now we can rewrite the fraction as [tex]\frac{8.88}{1.68}[/tex], and divide to get the final answer, which is 8.88 ÷ 1.68 ≈ 5.29. So the answer to the expression above is approximately 5.29.
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Convert 9.335 • 10^5 to standard form.
Find the value of x.
X
127⁰
40°
a. 90°
b. 95°
c. 100°
d. 103°
Answer:
d
Step-by-step explanation:
the sum of the interior angles in a quadrilateral = 360°
sum the angles and equate to 360°
x + 40° + 90° + 127° = 360°
x + 257° = 360° ( subtract 257° from both sides )
x = 103°
Show your work for multiplying the polynomials below and put your answer in standard form in the box below:
(x + 6) ( x ^2 - 3x - 4)
Multiplication of given polynomials = x³ + 3x² - 22x - 2
What is polynomial?A polynomial is a type of algebraic expression in which the exponents of all variables should be a whole number. The exponents of the variables in any polynomial have to be a non-negative integer.
Given,
(x + 6) ( x ^2 - 3x - 4)
Using distributive property
x( x ^2 - 3x - 4) + 6( x ^2 - 3x - 4)
Again using distributive property
x³ - 3x² - 4x + 6x² - 18x - 24
Adding like terms
x³ + 3x² - 22x - 2
Hence, x³ + 3x² - 22x - 24 is multiplication of given polynomials.
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Which of the following transformations is shown in the graph below?
Answer:
Red-Blue: Translate 2 units down, and translate 5 units right.
Blue-Red: Translate 2 units up, and translate 5 units left.
Step-by-step explanation:
Hi, I don't know if you want to go from red to blue or blue to red, but I'll answer both ways.
(2 points) conditional probability. a fair coin is tossed twice. (a) (1 point) what is the probability of two heads if we know that the first toss is a head? (b) (1 point) what is the probability of two heads if we know that at least one toss is a head?
If we know that the first toss is a head, the probability of two heads is 1/2. The probability of two heads if we know that at least one toss is a head is 2/3.
(a) If we know that the first toss is a head, the probability of two heads is 1/2 because the second toss is still independent and has a 50% chance of being either heads or tails.
(b) If we know that at least one toss is a head, the probability of two heads can be found by adding up the probability of getting two heads on the first and second tosses and then getting heads on the first and tails on the second toss. This can be represented as:
P(two heads | at least one head) = P(two heads) / P(at least one head) = (P(heads on first and heads on second) + P(heads on first and tails on second)) / (1 - P(tails on first and tails on second))
Since each toss has a 50% chance of being heads, the probability of getting two heads on the first and second tosses is 0.5 × 0.5 = 0.25. The probability of getting heads on the first and tails on the second is also 0.5 × 0.5 = 0.25. And the probability of getting tails on both tosses is 0.5 × 0.5 = 0.25.
Therefore, the answer is:
P(two heads | at least one head) = (0.25 + 0.25) / (1 - 0.25) = 0.5 / 0.75 = 2/3
So the probability of two heads, if we know that at least one toss is a head, is 2/3.
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Emilia and Miranda are sisters and their mother just signed them up for a new cell phone plan because they send too many text messengers Using I formation below detriment which sister sends the most text messages how many more text messengers does this sister send per week
to test the effect of a new drug on depression, we randomly assign people to control and experimental groups. those in the control group take a pill that contains no medication. this is a .
This is a Placebo-controlled study .A form of experimental design used in medical research to evaluate the efficacy of a novel medication or treatment is the placebo-controlled study.
Participants in the study are divided into two groups: the experimental group receives the new medicine under study, while the control group receives a placebo (the control group). The placebo is typically an inert chemical that mimics the real drug but has no therapeutic benefit, like a sugar pill
In this kind of study, the use of a control group is done to separate the effects of the medicine under test from other variables that might affect the results.
Researchers can compare the outcomes of the two groups to ascertain whether the new medication actually treats the illness of interest or whether any reported effects are caused by the placebo effect, other variables, or the disease's natural development.
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solve the following systems of linear equations using substitution
4x - 2y = 20 and x - 4y = -16
Answer:
The solution to this system of linear equations is (x, y) = (8, 6)
Step-by-step explanation:
We can solve this system of linear equations using substitution. To start, we'll isolate one of the variables in one of the equations, then substitute that expression into the other equation.
Starting with the first equation, we'll isolate x:
4x - 2y = 20
4x = 20 + 2y
x = 5 + 0.5y
Next, we'll substitute this expression for x into the second equation:
x - 4y = -16
5 + 0.5y - 4y = -16
5 - 3.5y = -16
Adding 3.5y to both sides:
5 = 3.5y - 16
Adding 16 to both sides:
21 = 3.5y
Finally, dividing both sides by 3.5:
y = 6
So, the value of y is 6. To find x, we can substitute this value back into the expression we found earlier:
x = 5 + 0.5y
x = 5 + 0.5(6)
x = 5 + 3
x = 8
So, the solution to this system of linear equations is (x, y) = (8, 6).
Answer:
(8, 6)
Step-by-step explanation:
Given system of linear equations:
[tex]\begin{cases}4x - 2y = 20 \\x - 4y = -16\end{cases}[/tex]
Rearrange the second equation to isolate x:
[tex]\implies x=4y-16[/tex]
Substitute the found expression for x into the first equation and solve for y:
[tex]\implies 4(4y-16)-2y=20[/tex]
[tex]\implies 16y-64-2y=20[/tex]
[tex]\implies 14y-64=20[/tex]
[tex]\implies 14y=84[/tex]
[tex]\implies y=6[/tex]
Substitute the found value of y into the expression for x and solve for x:
[tex]\implies x=4(6)-16[/tex]
[tex]\implies x=24-16[/tex]
[tex]\implies x=8[/tex]
Therefore, the solution to the given system of linear equations is (8, 6).
Brianna wants to buy a digital camera for a photography class. One store offers the camera for $60 down and a payment plan of $10 per month. The payment plan for a second store is described by y 20x + 60, where y is the total cost in dollars and x is the number of months. Which camera is cheaper when the camera is paid off in 12 months? Explain.
Please show your work! I will give brainliest answer quick
The cost of the camera in the first store is cheaper as the downpayment is the same for both stores but the monthly payment is cheaper in the first store.
What is a linear equation?An equation of degree one is known as a linear equation.
A linear equation of two variables can be represented by ax + by = c.
The total cost for the first store is,
y = 10(12) + 60.
y = 120 + 60.
y = $160.
The total cost for the second store is,
y = 20(12) + 60.
y = 240 + 60.
y = $300.
So, The first store is cheaper.
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Find the value of xx, yy, and zz, in the rhombus below.
Answer:
x = 6 , y = 30 , z = 66
Step-by-step explanation:
• All 4 sides of a rhombus are congruent
given one side is 59 , then all 4 sides equal 59
10x - 1 = 59 ( add 1 to both sides )
10x = 60 ( divide both sides by 10 )
x = 6
-------------------
2y - 1 = 59 ( add 1 to both sides )
2y = 60 ( divide both sides by 2 )
y = 30
--------------------
z - 7 = 59 ( add 7 to both sides )
z = 66
An electrician leans an extension ladder against the outside wall of a house so that it reaches an electric box 29 feet up. The ladder makes an angle of 71∘
with the ground. Find the length of the ladder. Round your answer to the nearest tenth of a foot if necessary.
Answer:
30.7
Step-by-step explanation:
sin 71° = 29/ ladder length
so length = 29/ sin 71° = 30.7
3. Find the sum, if it exists, of the infinite geometric series S = 21+9+3. 86 +.
O S = 36. 75
=
This infinite geometric series diverges.
O S = 12. 1
O S= 15. 75
36.75 is the sum of infinite geometric series .
What are geometric series that are finite and infinite?
The total of a finite set of terms is known as a finite series. An infinite series has an infinitely large term count and an infinitely high upper bound. Infinite series include, for instance, n=110(12)n1.
The sigma notation is followed by the infinity symbol, which denotes the endless nature of the series.
The formula for the sum of infinite geometric series is
S = a/1 - r
r - common ratio. |r|<1.
Here, a = first term = 21.
r - common ratio = 0.4285
Substituting the values in the formula, we get,
s = a/1 - r
= 21/(1 - 0.4285 ) = 21/0.5715
= 36.75
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A gardener wants to divide a square piece of lawn in half diagonally. what is the length of the diagonal side of the square is 8 ft ? leave your answer in simplest radical form.a. 16b.2c.8d. 4
The length of the diagonal side is 82, or answer (d) in its simplest radical form.
The Pythagorean Theorem must be utilized in order to determine the length of the diagonal side of a square with a length of 8 feet. You can use the formula a2 + b2 = c2, where a and b are the sides of the triangle and c is the hypotenuse, since the diagonal side of the square is. Because a and b are both 8 feet, you can use those numbers to solve for c:
82 + 82 = c2 64 + 64 = c2 128 = c2 The length of the diagonal side is 82, or answer d in its simplest radical form.
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